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RANDOM WALK GENERATOR

## This function
## provides us with very simple
## random walk generator compared
## to the function rand_disc_loop
## which is time-consuming, in operation,
## long, and unuseful for other than the
## coin toss simulatons
function xy=rand_disc(N); ## RW generator.
r=floor(rand(N,1)*4); ## Random coulumn vector of N
## integer elements
## multiplied by 4 to widen the
## interval from [0,1] to [0,3].
x=y=zeros(size(r)); ## The coulumn vectors
## of N elements with all elements zero.
x(find(r==0)) = 1; ## The elements of x,
## the vector function
## which takes in the row #
## of the zero elements of
## the vector r as an argument,
## is assigned to 1.
x(find(r==1)) =-1; ## The elements of x,
## the vector function
## which takes in the row # of the
## elements of 1 of the vector r
## as an argument,
## is assigned to -1.
y(find(r==2)) = 1; ## The elements of y,
## the vector function
## which takes in the row # of the
## elements of 2 of the vector r
## as an argument,
## is assigned to 1.
y(find(r==3)) =-1; ## The elements of y,
## the vector function
## which takes in the row # of the
## elements of 3 of the vector r
## as an argument,
## is assigned to -1.
xy=[x y]; ## The resulting x by y
## matrix of a random walk
## with elements 1 and -1.
endfunction

## Newton-Rapson Method to the smallest non negative root
## of the 8th degree Legendre Polynomial
## P8(x)=(1/128)(6435x^8-12012x^6+6930x^4-1260x^2+35)
## where -1<=x<=1.
## for the smallest non negative root, we can ignore
## all the terms except the last two by truncated
## the function to be zero and find
## x=0.167 as the initial smallest non negative
## root.
##Constants and initializations
x=[]; ## Empty array for the iterated x roots
x(1)=0.16700000; ## Initial guess to begin the iteration for the
## smallest non-negative root.
L8=[]; ## Empty array for the Legendre polynomial
L8p=[]; ## Empty array for the derivative of the Legendre polynomial
for i=1:100
##The value of the function at x
L8(i)=(1/128)*(6435*x(i)^8-12012*x(i)^6+6930*x(i)^4-1260*x(i)^2+35);
##The value of the derivative of the function at x
L8p(i)=(1/128)*(6435*8*x(i)^7-12012*6*x(i)^5+6930*4*x(i)^3-1260*2*x(i));
x(i+1)=x(i)-L8(i)/L8p(i); ## the iteration
endfor
## For plot let's define a new variable…